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T_{UD}

Last-modified: 2013-04-19 (金) 13:22:13

Definition Edit

  • A topological space (X,τ) is said to be T_{UD} if for every point x of X, {x}' is a union of disjoint closed set (where {x}' is the derived set of {x}).

Property Edit

  • A topological space (X,τ) is a T_{UD} space iff ∀x,y∈X, y∈{x}' and {x}' is not a union of disjoint closed sets implies imgtex.fcgi?%5bres=100%5d%7b%5c%5b%20%5c%7by%5c%7d'=%5cemptyset%20%5c%5d%7d%25.png. [3]
  • T_{UD} = T_0 + R_{UD}. [2]
  • T_{UD} = T_R + R*_{UD}. [4]

Reference Edit

  1. Aull, Charles E.; Thron, W.J.,Separation axioms between T0 and T1., (English) [J] Nederl. Akad. Wet., Proc., Ser. A 65, 26-37 (1962).
  2. Misra, D. N.; Dube, K. K., Some axioms weaker than the R0-axiom., (Serbo-Croatian summary), Glasnik Mat. Ser. III 8(28) (1973), 145-148.
  3. Guia, Josep, Axioms weaker than R0., (Serbo-Croatian summary), Mat. Vesnik 36 (1984), no. 3, 195–205.
  4. Guia, Josep, Essentially T_D and essentially T_UD spaces., Bull. Math. Soc. Sci. Math. R. S. Roumanie (N.S.) 32(80) (1988), no. 3, 227-233.